Deep EvoGraphNet Architecture ForTime-Dependent Brain Graph Data Synthesis

From a Single Timepoint

Ahmed Nebli† ID 1,2, Ugur Ali Kaplan†1, and Islem Rekik ID 1?

1 BASIRA Lab, Faculty of Computer and Informatics, Istanbul Technical University,Istanbul, Turkey

2 National School for Computer Science (ENSI), Mannouba, Tunisia

Abstract. Learning how to predict the brain connectome (i.e. graph)development and aging is of paramount importance for charting the fu-ture of within-disorder and cross-disorder landscape of brain dysconnec-tivity evolution. Indeed, predicting the longitudinal (i.e., time-dependent)brain dysconnectivity as it emerges and evolves over time from a singletimepoint can help design personalized treatments for disordered pa-tients in a very early stage. Despite its significance, evolution modelsof the brain graph are largely overlooked in the literature. Here, wepropose EvoGraphNet, the first end-to-end geometric deep learning-powered graph-generative adversarial network (gGAN) for predictingtime-dependent brain graph evolution from a single timepoint. Our Evo-GraphNet architecture cascades a set of time-dependent gGANs, whereeach gGAN communicates its predicted brain graphs at a particular time-point to train the next gGAN in the cascade at follow-up timepoint.Therefore, we obtain each next predicted timepoint by setting the outputof each generator as the input of its successor which enables us to predicta given number of timepoints using only one single timepoint in an end-to-end fashion. At each timepoint, to better align the distribution of thepredicted brain graphs with that of the ground-truth graphs, we furtherintegrate an auxiliary Kullback-Leibler divergence loss function. To cap-ture time-dependency between two consecutive observations, we imposean l1 loss to minimize the sparse distance between two serialized braingraphs. A series of benchmarks against variants and ablated versions ofour EvoGraphNet showed that we can achieve the lowest brain graph evo-lution prediction error using a single baseline timepoint. Our EvoGraph-Net code is available at http://github.com/basiralab/EvoGraphNet.

Keywords: Time-dependent graph evolution prediction ·KL divergenceloss · graph generative adversarial network · cascaded time-dependentgenerators

? corresponding author: irekik@itu.edu.tr, http://basira-lab.com; †: co-first au-thors. This work is accepted for publication in the PRedictive Intelligence inMEdicine (PRIME) workshop Springer proceedings in conjunction with MICCAI2020.

arX

iv:2

009.

1321

7v1

[ee

ss.I

V]

28

Sep

2020

1 Introduction

Recent findings in neuroscience have suggested that providing personalized treat-ments for brain diseases can significantly increase the chance of a patient’s re-covery [1]. Thereby, it is vital to undertake an early diagnosis of brain diseases[2], especially for neurodegenerative diseases such as dementia which was foundto be irreversible if discovered at a late stage [3]. In this context, recent land-mark studies [4,5] have suggested using the robust predictive abilities of machinelearning to predict the time-dependent (i.e., longitudinal) evolution of both thehealthy and the disordered brain. However, such works only focus on predict-ing the brain when modeled as an image or a surface, thereby overlooking awide spectrum of brain dysconnectivity disorders that can be pinned down bymodeling the brain as a graph (also called connectome) [6], where the connec-tivity weight between pairs of anatomical regions of interest (ROIs) becomes thefeature of interest.

In other works, [7,8] have proposed to use deep learning frameworks aiming toutilize hippocampal magnetic resonance imaging (MRI) to predict the onset ofAlzheimer’s disease (AD). However, these studies have only focused on a singlebrain region overlooking other brain regions’ engagement in predicting AD. Toovercome this limitation, [9,10] proposed to use brain MRI to predict AD acrossall brain regions, yet such experiments focused on MRI samples collected atthe late stages of the illness which are inadequate for administering patient’spersonalized treatments at baseline observation. Here, we set out to solve a moredifficult problem which is forecasting the evolution of the brain graph over timefrom an initial timepoint. Despite the lack of studies in this direction, [11] hasattempted to solve this challenge by suggesting the Learning-guided InfiniteNetwork Atlas selection (LINAs) framework, a first-ever study that developed alearning-based sample similarity learning framework to forecast the progressionof brain disease development over time from a single observation only. This studyreflects a major advancement in brain disease evolution trajectory predictionsince it considered using the brain as a set of interconnected ROIs, instead ofperforming image-based prediction which is agnostic to the nature of the complexwiring of the brain as a graph.

Regardless of the fact that the aforementioned techniques were aiming toaddress the issue of early diagnosis of brain diseases, all of these approachesshare the same dichotomized aspect of the engineered learning framework, eachcomposed of independent blocks that cannot co-learn together to better solvethe target prediction problem. For instance, [11] first learns the data manifold,then learns how to select the best samples before performing the prediction stepindependently. These sub-components of the framework do not interact witheach other and thus there is no feedback passed on to earlier learning blocks.

To address these limitations and while drawing inspiration from the com-pelling generative adversarial network (GAN) architecture introduced in [12],we propose EvoGraphNet, a framework that first generalizes the GAN architec-ture, originally operating on Euclidean data such as images, to non-Euclideangraphs by designing a graph-based GAN (gGAN) architecture. Second and more

2

importantly, our EvoGraphNet chains a set of gGANs, each specialized in learn-ing how to generate a brain graph at follow-up timepoint from the predictedbrain graph (output) of the previous gGAN in the time-dependent cascade. Weformalize our brain graph prediction task from baseline using a single loss func-tion to optimize with a backpropagation process throughout our EvoGraphNetarchitecture, trained in an end-to-end fashion. Our framework is inspired by theworks of [13,14] where we aim to perform an assumption free mapping froman initial brain graph to its consecutive time-dependent representations using astack of m paired generators and discriminators at m follow-up timepoints. Eachgenerator inputs the output of its predecessor generator making the frameworkwork in an end-to-end fashion. We further propose to enhance the quality ofthe evolved brain graphs from baseline by maximizing the alignment betweenthe predicted and ground-truth brain graphs at each prediction timepoint byintegrating a Kullback-Leibler (KL) divergence. To capture time-dependencybetween two consecutive observations, we impose an l1 loss to minimize thesparse distance between two serialized brain graphs. We also explore the effectof adding a graph-topology loss where we enforce the preservation of the topo-logical strength of each ROI (graph node) strength over time.

Below, we articulate the main contributions of our work:

1. On a conceptual level. EvoGraphNet is the first geometric deep learningframework that predicts the time-dependent brain graph evolution from asingle observation in an end-to-end fashion.

2. On a methodological level. EvoGraphNet is a unified prediction frameworkstacking a set of time-dependent graph GANs where the learning of thecurrent gGAN in the cascade benefits from the learning of the previousgGAN.

3. On clinical level. EvoGraphNet can be used in the development of a morepersonalized medicine for the early diagnosis of brain dysconnectivity disor-ders.

2 Proposed Method

In this section, we explain the key building blocks of our proposed EvoGraph-Net architecture for time-dependent brain graph evolution synthesis from a singlebaseline timepoint. Table 1 displays the mathematical notations used through-out our paper.

Overview of EvoGraphNet for time-dependent brain graph evo-lution prediction from a single source timepoint. GANs [15] are deeplearning models consisting of two neural networks competing in solving a targetlearning task: a generator G and a discriminator D. The generator is an encoderand decoder neural network aiming to learn how to generate fake samples thatmimics a given original data distribution while the discriminator learns how to

3

Fig

.1:Proposed

EvoGraphNet

arch

itecture

forpred

ictingthelongitu

dinalbra

ingra

phevo

lutio

nfro

mbaselin

etim

epointt0 .

(A)

Stacked

genera

tors

netw

ork.

We

dev

elopa

graph

GA

N(g

GA

N)

that

learn

sh

owto

map

an

inp

ut

brain

graph

measu

redatti−

1

toa

follow-u

ptim

epoin

tti .

Form

timep

oints,

we

desig

nm

iden

tical

gra

ph

gen

erato

rsea

chcom

posed

ofa

three-layer

graph

convo

lutio

nal

neu

ralnetw

orkactin

gas

anen

cod

er-deco

der

netw

ork

that

mim

icsa

U-n

etarch

itecture

with

skip

conn

ections.

For

eachti ,i>

0,ea

chgen

eratorG

iin

the

chain

takes

aset

of

pred

ictedtra

inin

gb

rain

gra

ph

sX

trti−

1by

the

prev

ious

generator

an

dou

tpu

tsa

setof

Xti .

(B)Stacked

discrim

inatornetw

ork.

For

each

gen

erato

rG

i ,w

easso

ciatea

discrim

inator

Di

aimin

g

tod

istingu

ishth

egrou

nd

-truth

bra

ingrap

hs

Xtrti

from

the

pred

ictedb

rain

gra

ph

sX

trti

by

generator

Gi .

Hen

ce,w

ed

esignm

iden

tical

discrim

inators

eachcom

posed

ofa

two-layer

gra

ph

convo

lutio

nal

neu

ral

netw

ork

,each

outp

uttin

ga

single

scorein

the

range

of

[0,...,1]

exp

ressing

therea

lness

of

Xtrti .

To

enh

an

ceth

egen

eratio

nqu

ality

at

eachfollow

-up

timep

ointti ,

we

furth

erin

tegratea

Ku

llback

-Leib

lerd

ivergence

loss

tob

etteralig

nth

ed

istribu

tion

of

the

pred

ictedb

raingrap

hs

with

that

ofth

egro

un

d-tru

thgra

ph

s.

4

Table 1: Major mathematical notationsMathematical notation Definition

m number of timepoints to predictns number of training subjectsnr number of regions of interest in the brainmr number of edges in a brain graphµk mean of the connected edge weights for node k in a ground-truth brain graphσk standard deviation of the connected edge weights for node k in a ground-truth brain graphµk mean of the connected edge weights for node k in a predicted brain graphσk standard deviation of the connected edge weights for node k in a predicted brain graphpk normal distribution of the connected edge weights for a node k in a ground-truth brain graphpk normal distribution of the connected edge weights for a node k in the ground-truth brain graphXtr

ti training brain graph connectivity matrices ∈ Rn×nr×nr at tiXtr

ti predicted brain graph connectivity matrices ∈ Rn×nr×nr at tiGi gGAN generator at timepoint tiDi gGAN discriminator at timepoint tiLfull full loss functionLadv adversarial loss functionLL1 l1 loss functionLKL KL divergence loss functionλ1 coefficient of adversarial lossλ2 coefficient of l1 lossλ3 coefficient of KL divergence lossV a set of nr nodesE a set of mr directed or undirected edgesl index of layerL transformation matrix ∈ Rmr×s

N (k) the neighborhood containing all the adjacent nodes of node k

Yl(k) filtered signal of node (ROI) i ∈ Rdl

Flk′k filter generating network

ωl weight parameter

bl bias parameter

discriminate between the ground-truth data and the fake data produced by thegenerator. These two networks are trained in an adversarial way so that withenough training cycles, the generator learns how to better generate fake samplesthat look real and the discriminator learns how to better differentiate betweenground-truth samples and generator’s produced samples. We draw inspirationfrom the work of [16] which successfully translated a T1-weighted magnetic res-onance imaging (MRI) to T2-weighted MRI using GAN and the work of [17]which proposed to use a stacked form of generators and discriminators for imagesynthesis. Therefore, as shown in Fig. 1 our proposed EvoGraphNet is a chain ofgGANs composed of m generators and discriminators aiming to map a subject’sbrain graph measured at timepoint t0 onto m sequential follow-up timepoints.Excluding the baseline gGAN trained to map ground-truth baseline graphs at t0to the ground-truth brain graphs at timepoint t1, a gGAN at time ti is trained tomap the generated graphs by the previous gGAN at ti−1 onto the ground-truthbrain graphs at timepoint ti. Below is our adversarial loss function for one pairof generator Gi and discriminator Di composing our gGAN at timepoint ti:

argminGimaxDi

Ladv(Gi, Di) = EGi(Xti)[log(Di(Gi(Xti))] + EGi(Xti−1

)[log(1−Di(Gi(Xti−1))]

(1)

Xti−1denotes the predicted brain graph connectivity matrix by the previous

generator Gi−1 in the chain, and which is inputted to the generator Gi for train-

5

ing. Xtrti denotes the target ground-truth brain connectivity matrix at timepoint

ti. Gi is trained to learn a non-linear mapping from the previous prediction Xti−1

to the target ground truth Xtrti .

Since the brain connectivity changes are anticipated to be sparse over time, wefurther impose that the l1 distance between two consecutive timepoints ti−1 andti to be quite small for i ∈ {1, . . . ,m}. This acts as a time-dependent regularizerfor each generator Gi in our EvoGraphNet architecture. Thus, we express theproposed l1 loss for each subject tr using the predicted brain graph connectivitymatrix Xtr

ti−1by the previous generator Gi−1 and the ground-truth brain graph

connectivity matrix to predict by generator Gi as follows:

Ll1(Gi, tr) = ||Xtrti−1−Xtr

ti ||1 (2)

In addition to the l1 loss term introduced for taking into account time-dependency between two consecutive predictions, we further enforce the align-ment between the ground-truth and predicted brain graph distribution at eachtimepoint ti. Precisely, we use the Kullback-Leibler (KL) divergence betweenboth ground-truth and predicted distributions. KL divergence is a metric in-dicating the ability to discriminate between two given distributions. Thereby,we propose to add the KL divergence as a loss term to minimize the discrep-ancy between ground-truth and predicted connectivity weight distributions ateach timepoint ti. To do so, first, we calculate the mean µk and the standarddeviation σk of connected edge weights for each node in each brain graph.

We define the normal distributions as follows:

pk = N(µk, σk) (3)

qk = N(µk, σk), (4)

where pk is the normal distribution for node k in the predicted graph Xtrti =

Gi(Xtrti−1

) and qk is the normal distribution defined for the same node k in Xtrti .

Thus, the KL divergence between the previously calculated normal distribu-tions for each subject is expressed as follows:

LKL(ti, tr) =

nr∑k=1

KL(pk||qk), (5)

where each node’s KL divergence is equal to:

KL(pk||qk) =

∫ +∞

−∞pk(x)log

pk(x)

qk(x)dx (6)

6

Full loss. By summing up over all training ns subjects and all m timepointsto predict, we obtain the full loss function to optimize follows:

LFull =

m∑i=1

(λ1LAdv(Gi, Di) +

λ2ns

ns∑tr=1

Ll1(Gi, tr) +λ3ns

ns∑tr=1

LKL(ti, tr)

)(7)

where λ1, λ2, and λ3 are hyperparameters controlling the significance of eachloss function in the overall objective to minimize.

The generator network design. As shown in Fig. 1–A, our proposed Evo-GraphNet is composed m generators aiming to predict the subject’s brain graphsat m follow-up timepoints. Since all generators are designed identically and forthe sake of simplicity, we propose to detail the architecture of one generator Gi

aiming to predict subjects’ brain graphs at timepoint ti.

Our proposed generator Gi consists of a three-layer encoder-decoder graphconvolutional neural network (GCN) leveraging the dynamic edge convolutionprocess implemented in [18] and imitating a U-net architecture [19] with skipconnections which enhances the decoding process with respect to the encoder’sembedding. For each ti, i > 0, each generator Gi in the chain takes a set ofpredicted training brain graphs Xtr

ti−1by the previous generator and outputs a

set of Xti . Hence, our generator Gi contains three graph convolutional neuralnetwork layers to which we apply batch normalization [20] and dropout [21] tothe output of each layer. These two operations undeniably contribute to sim-plifying and optimizing the network training. For instance, batch normalizationwas proven to accelerate network training while dropout was proven to eliminatethe risk of overfitting.

The discriminator network design. We display the architecture of ourdiscriminator in Fig. 1–B. Similar to the above sub-section, all discriminatorsshare the same design, thus we propose to only detail one discriminator Di

trained at timepoint ti. We couple each generator Gi with its correspondingdiscriminator Di which is also a graph neural network inspired by [18]. Ourproposed discriminator is a two-layer graph neural network that takes as inputa concatenation of the generator’s output Xtr

ti and the ground-truth subjects’brain graphs Xtr

ti . The discriminator outputs a value in the range of [0, . . . , 1]

measuring the realness of the synthesized brain graph Xtrti at timepoint ti. As

reflected by our adversarial loss function, we design our gGAN’s loss function sothat it maximizes the discriminator’s output score for Xtr

ti and minimizes it for

Xtrti , which is the output of the previous generator in the chain Gi(X

trti−1

).

Dynamic graph-based edge convolution. Each of the graph convolu-tional layers of our gGAN architecture uses a dynamic graph-based edge convo-lution operation proposed by [18]. In particular, let G = (V,E) be a directed orundirected graph where V is a set of nr ROIs and E ⊆ V × V is a set of mr

edges. Let l be the layer index in the neural network. We define Yl : V → Rdl

7

and L : E → Rdm which can be respectively considered as two transforma-tion matrices (i.e., functions) where Yl ∈ Rnr×dl and L ∈ Rmr×dm . dm anddl are dimensionality indexes. We define by N (k) = {k′; (k′, k) ∈ E} ∪ {k} theneighborhood of a node k containing all its adjacent ROIs.

The goal of each layer in each generator and discriminator in our EvoGraph-Net is to output the graph convolution result which can be considered as afiltered signal Yl(k) ∈ Rdl at node k’s neighborhood k′ ∈ N (k). Yl is expressedas follows:

Yl(k) =1

N (k)

∑k′∈N (k)

Θlk′kYl−1(k′) + bl, (8)

where Θlk′k = Fl(L(k′, k);ωl). We note that Fl : Rdm → Rdl×dl−1 is the

filter generating network, ωl and bl are model parameters that are updated onlyduring training.

3 Results and Discussion

Evaluation dataset. We used 113 subjects from the OASIS-21 longitudinaldataset [22]. This set consists of a longitudinal collection of 150 subjects aged 60to 96. Each subject has 3 visits (i.e., timepoints), separated by at least one year.For each subject, we construct a cortical morphological network derived fromcortical thickness measure using structural T1-w MRI as proposed in [23,24,25].Each cortical hemisphere is parcellated into 35 ROIs using Desikan-Killiany cor-tical atlas. We program our EvoGraphNet using PyTorch Geometric library [26].

Parameter setting. In Table 2, we report the mean absolute error betweenground-truth and synthesized brain graphs at follow-up timepoints t1 and t2.We set each pair of gGAN’s hyperparameter as follows: λ1 = 2, λ2 = 2, andλ3 = 0.001. Also, we chose AdamW [27] as our default optimizer and set thelearning rate at 0.01 for each generator and 0.0002 for each discriminator. Weset the exponential decay rate for the first moment estimates to 0.5, and theexponential decay rate for the second-moment estimates to 0.999 for the AdamWoptimizer. Finally, we trained our gGAN for 500 epochs using a single Tesla V100GPU (NVIDIA GeForce GTX TITAN with 32GB memory).

Comparison Method and evaluation. To evaluate the reproducibilityof our results and the robustness of our EvoGraphNet architecture to trainingand testing sample perturbation, we used a 3-fold cross-validation strategy fortraining and testing. Due to the lack of existing works on brain graph evolutionprediction using geometric deep learning, we proposed to evaluate our methodagainst two of its variants. The first comparison method (i.e., base EvoGraphNet)is an ablated of our proposed framework where we remove the KL divergenceloss term while for the second comparison method (i.e., EvoGraphNet (w/oKL) + Topology), we replace our KL-divergence loss with a graph topology

1 https://www.oasis-brains.org/

8

loss. Basically, we represent each graph by its node strength vector storing thetopological strength of each of its nodes. The node strength is computed byadding up the weights of all edges connected to the node of interest. Next, wecomputed the L2 distance between the node strength vector of the ground-truthand predicted graphs. Table 2 shows the MAE results at t1 and t2 timepoints.

Table 2: Prediction accuracy using mean absolute error (MAE) of our proposedmethod and comparison methods at t1 and t2 timepoints.

t1 t2

MethodMean MAE± std

BestMAE

Mean MAE± std

BestMAE

Base EvoGraphNet (w/o KL) 0.05626± 0.00446 0.05080 0.13379± 0.01385 0.11586EvoGraphNet (w/o KL) + Topology 0.05643± 0.00307 0.05286 0.11194± 0.00381 0.10799

EvoGraphNet 0.05495± 0.00282 0.05096 0.08048± 0.00554 0.07286

Clearly, our proposed EvoGraphNet outperformed the comparison methodsin terms of mean (averaged across the 3 folds) and the best MAE for predictionat t2. It has also achieved the best mean MAE for brain graph prediction at t1.However, the best MAE for prediction at t1 was achieved by base EvoGraphNetwhich used only a pair of loss functions (e.i., adversarial loss and l1 loss). Thismight be due to the fact that our objective function is highly computationallyexpensive compared to the base EvoGraphNet making it less prone to learningflaws such as overfitting. Also, we notice that the prediction error at t1 is lowerthan t2, which indicates that further observations become more challenging topredict from the baseline timepoint t0. Overall, our EvoGraphNet architecturewith an additional KL divergence loss has achieved the best performance inforeseeing brain graph evolution trajectory and showed that stacking a pairof graph generator and discriminator for each predicted timepoint is indeed apromising strategy in tackling our time-dependent graph prediction problem.

Limitations and future work. While our graph prediction frameworkreached the lowest average MAE against benchmark methods in predicting theevolution of brain graphs over time from a single observation, it has a few limita-tions. So far, the proposed method only handles brain graphs where a single edgeconnects two ROIs. In our future work, we aim to generalize our stacked gGANgenerators to handle brain hypergraphs, where a hyperedge captures high-orderrelationships between sets of nodes. This will enable us to better model and cap-ture the complexity of the brain as a highly interactive network with differenttopological properties. Furthermore, we noticed that the joint integration of KLdivergence loss and the topological loss produced a negligible improvement in thebrain graph evolution prediction over time. We intend to investigate this pointin depth in our future work. Besides, we aim to use a population network atlasas introduced in [28] to assist EvoGraphNet training in generating biologicallysound brain networks.

9

4 Conclusion

In this paper, we proposed a first-ever geometric deep learning architecture,namely EvoGraphNet, for predicting the longitudinal evolution of a baselinebrain graph over time. Our architecture chains a set of gGANs where the learn-ing of each gGAN in the chain depends on the output of its antecedent gGAN.We proposed a time-dependency loss between consecutive timepoints and a dis-tribution alignment between predicted and ground-truth graphs at the sametimepoint. Our results showed that our time-dependent brain graph generationframework from the baseline timepoint can notably boost the prediction accu-racy compared to its ablated versions. Our EvoGraphNet is generic and can betrained using any given number of prediction timepoints, and thus can be used inpredicting both typical and disordered changes in brain connectivity over time.Therefore, in our future research, we will be testing our designed architecture onlarge-scale connectomic datasets of multiple brain disorders such as schizophre-nia by exploring the ability of the synthesized brain graphs at later timepointsto improve neurological and neuropsychiatric disorder diagnosis from baseline.

5 Supplementary material

We provide three supplementary items for reproducible and open science:

1. A 6-mn YouTube video explaining how our prediction framework works onBASIRA YouTube channel at https://youtu.be/aT---t2OBO0.

2. EvoGraphNet code in Python on GitHub at https://github.com/basiralab/EvoGraphNet.

3. A GitHub video code demo of EvoGraphNet on BASIRA YouTube channelat https://youtu.be/eTUeQ15FeRc.

6 Acknowledgement

This project has been funded by the 2232 International Fellowship for Out-standing Researchers Program of TUBITAK (Project No:118C288, http://

basira-lab.com/reprime/) supporting I. Rekik. However, all scientific con-tributions made in this project are owned and approved solely by the authors.

References

1. Mukherjee, A., Srivastava, R., Bhatia, V., Mohanty, S., et al.: Stimuli effect ofthe human brain using eeg spm dataset. In: Smart Healthcare Analytics in IoTEnabled Environment. Springer (2020) 213–226

2. Lohmeyer, J.L., Alpinar-Sencan, Z., Schicktanz, S.: Attitudes towards predictionand early diagnosis of late-onset dementia: a comparison of tested persons andfamily caregivers. Aging & Mental Health (2020) 1–12

10

3. Stoessl, A.J.: Neuroimaging in the early diagnosis of neurodegenerative disease.Translational Neurodegeneration 1 (2012) 5

4. Rekik, I., Li, G., Yap, P., Chen, G., Lin, W., Shen, D.: Joint prediction of longitu-dinal development of cortical surfaces and white matter fibers from neonatal MRI.Neuroimage 152 (2017) 411–424

5. Gafuroglu, C., Rekik, I., et al.: Joint prediction and classification of brain imageevolution trajectories from baseline brain image with application to early dementia.International Conference on Medical Image Computing and Computer-AssistedIntervention (2018) 437–445

6. van den Heuvel, M.P., Sporns, O.: A cross-disorder connectome landscape of braindysconnectivity. Nature reviews neuroscience 20 (2019) 435–446

7. Li, H., Habes, M., Wolk, D.A., Fan, Y.: A deep learning model for early predic-tion of Alzheimer’sdisease dementia based on hippocampal mri. arXiv preprintarXiv:1904.07282 (2019)

8. Liu, M., Li, F., Yan, H., Wang, K., Ma, Y., Shen, L., Xu, M., Initiative, A.N.,et al.: A multi-model deep convolutional neural network for automatic hippocam-pus segmentation and classification in Alzheimer’sdisease. NeuroImage 208 (2020)116459

9. Zhang, Y., Dong, Z., Phillips, P., Wang, S., Ji, G., Yang, J., Yuan, T.F.: Detectionof subjects and brain regions related to Alzheimer’sdisease using 3d mri scans basedon eigenbrain and machine learning. Frontiers in computational neuroscience 9(2015) 66

10. Islam, J., Zhang, Y.: A novel deep learning based multi-class classification methodfor Alzheimer’sdisease detection using brain mri data. In: International Conferenceon Brain Informatics, Springer (2017) 213–222

11. Ezzine, B.E., Rekik, I.: Learning-Guided Infinite Network Atlas Selection for Pre-dicting Longitudinal Brain Network Evolution from a Single Observation. In Shen,D., Liu, T., Peters, T.M., Staib, L.H., Essert, C., Zhou, S., Yap, P.T., Khan, A.,eds.: Medical Image Computing and Computer Assisted Intervention MICCAI2019. Volume 11765. Springer International Publishing, Cham (2019) 796–805

12. Goodfellow, I.J., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair,S., Courville, A., Bengio, Y.: Generative adversarial networks (2014)

13. Isola, P., Zhu, J., Zhou, T., Efros, A.A.: Image-to-image translation with condi-tional adversarial networks. CoRR abs/1611.07004 (2016)

14. Welander, P., Karlsson, S., Eklund, A.: Generative adversarial networks for image-to-image translation on multi-contrast mr images-a comparison of cyclegan andunit. arXiv preprint arXiv:1806.07777 (2018)

15. Goodfellow, I., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair, S.,Courville, A., Bengio, Y.: Generative adversarial nets. In Ghahramani, Z., Welling,M., Cortes, C., Lawrence, N.D., Weinberger, K.Q., eds.: Advances in Neural Infor-mation Processing Systems 27. Curran Associates, Inc. (2014) 2672–2680

16. Yang, Q., Li, N., Zhao, Z., Fan, X., Eric, I., Chang, C., Xu, Y.: Mri cross-modalityimage-to-image translation. Scientific Reports 10 (2020) 1–18

17. Zhang, H., Xu, T., Li, H., Zhang, S., Huang, X., Wang, X., Metaxas, D.N.: Stack-gan: Text to photo-realistic image synthesis with stacked generative adversarialnetworks. CoRR abs/1612.03242 (2016)

18. Simonovsky, M., Komodakis, N.: Dynamic edge-conditioned filters in convolutionalneural networks on graphs. CoRR abs/1704.02901 (2017)

19. Ronneberger, O., Fischer, P., Brox, T.: U-net: Convolutional networks for biomed-ical image segmentation. CoRR abs/1505.04597 (2015)

11

20. Ioffe, S., Szegedy, C.: Batch normalization: Accelerating deep network training byreducing internal covariate shift. arXiv preprint arXiv:1502.03167 (2015)

21. Xiao, T., Li, H., Ouyang, W., Wang, X.: Learning deep feature representationswith domain guided dropout for person re-identification. In: Proceedings of theIEEE conference on computer vision and pattern recognition. (2016) 1249–1258

22. Marcus, D.S., Fotenos, A.F., Csernansky, J.G., Morris, J.C., Buckner, R.L.: Openaccess series of imaging studies: longitudinal mri data in nondemented and de-mented older adults. Journal of cognitive neuroscience 22 (2010) 2677–2684

23. Mahjoub, I., Mahjoub, M.A., Rekik, I.: Brain multiplexes reveal morphologicalconnectional biomarkers fingerprinting late brain dementia states. Scientific reports8 (2018) 4103

24. Soussia, M., Rekik, I.: Unsupervised manifold learning using high-order morpho-logical brain networks derived from T1-w MRI for autism diagnosis. Frontiers inNeuroinformatics 12 (2018)

25. Lisowska, A., Rekik, I., ADNI: Pairing-based ensemble classifier learning usingconvolutional brain multiplexes and multi-view brain networks for early dementiadiagnosis. International Workshop on Connectomics in Neuroimaging (2017) 42–50

26. Fey, M., Lenssen, J.E.: Fast graph representation learning with pytorch geometric.CoRR abs/1903.02428 (2019)

27. Loshchilov, I., Hutter, F.: Fixing weight decay regularization in adam. CoRRabs/1711.05101 (2017)

28. Mhiri, I., Rekik, I.: Joint functional brain network atlas estimation and featureselection for neurological disorder diagnosis with application to autism. Medicalimage analysis 60 (2020) 101596

12